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Beginning Algebra
Tutorial 29: Negative Exponents and Scientific Notation

 

deskLearning Objectives

After completing this tutorial, you should be able to:
  1. Simplify exponential expressions involving negative exponents.
  2. Write a number in scientific notation.
  3. Write a number in standard notation, without exponents.




desk Introduction


This tutorial picks up where Tutorial 26: Exponents left off.  It finishes the rules of exponents with negative exponents.  Also we will go over scientific notation. Like it or not, the best way to master these exponents is to work through exponent problems.  So I guess we better get to it.

 

 

desk Tutorial



 

Negative Exponents
negative exponent   or negative exponent

 
Be careful with negative exponents.  The temptation is to negate the base, which would not be a correct thing to do. Since exponents are another way to write multiplication and the negative is in the exponent, to write it as a positive exponent we do the multiplicative inverse which is to take the reciprocal of the base.
 

notebook Example 1:  Simplify example 1a.

 
example 1b
*Rewrite with a pos. exp. by taking recip. of base 
 

*Use def. of exponents to evaluate

 
 
notebook Example 2:  Simplify example 2a.

 
example 2b
*Rewrite with a pos. exp. by taking recip. of base 

*Use def. of exponents to evaluate

 
 
Simplifying an Exponential Expression

 
When simplifying an exponential expression,  write it so that each base is written one time with one POSITIVE exponent. 

In other words, write it in the most condense form you can making sure that all your exponents are positive.

A lot of times you are having to use more than one rule to get the job done.  As long as you are using the rule appropriately, you should be fine. 

 
 
Review of Exponent Rules

 
Except for the negative exponent rule, examples of the following rules can be found in Tutorial 26: Exponents.
 

Product Rule: product

Power Rule for Exponents: power

Power of a Product: product

Power of a Quotient: power and quotient

Quotient Rule for Exponents: quotient

Zero Exponent: zero exponent
Negative Exponent: negative exponent
 

 
 
notebook Example 3:    Simplify.  Write answer with positive exponents.
example 3a

 
example 3b

*Rewrite with a pos. exp. by taking recip. of base 
 

 
 
notebook Example 4:   Simplify.  Use positive exponents to write the answer.
example 4a

 
example 4b

*Rewrite with a pos. exp. by taking recip. of base
 
 

 

 
 
notebook Example 5:   Simplify.  Use positive exponents to write the answer.
example 5a

 
 
example 5b

*Rewrite with a pos. exp. by taking recip. of base

 

 
 
notebook Example 6:    Simplify.  Write answer with positive exponents.
example 6a

 
example 6b
 

*When mult. like bases you add your exponents
 
 

*When div. like bases you subtract your exponents
 

*Rewrite with a pos. exp. by taking recip. of base 
 

 
Be careful going into the last line.  Note that you do not see an exponent written with the number 5.  This means that the exponent on 5  is understood to be 1.  Since it doesn't have a negative exponent, we DO NOT take the reciprocal of 5.  The only base that has a negative exponent is a, so a is the only base we take the reciprocal of.


 

Scientific Notation

A positive number is written in scientific notation if it is written in the form:

scientific

where 1 < a < 10 and r is an integer power of 10.

 
 
 
Writing a Number in Scientific Notation

 
Step 1:  Move the decimal point so that you have a number that is between 1 and 10.

 
In other words, you will put your decimal after the first non zero number. 

 
Step 2:   Count the number of decimal places moved in Step 1 .

 
If the decimal point was moved to the left, the count is positive.

If the decimal point is moved to the right, the count is negative.

 
Step 3:   Write as a product of the number (found in Step 1) and 10 raised to the power of the count (found in Step 2).

 
 
 
notebook Example 7:    Write the number in scientific notation:   483,000,000.

 

 
example 7a
*Decimal is at the end of the number

*Move decimal to create a number between 1 and 10

 

 
How many decimal places did we end up moving?
We started at the end of the number 483000000 and moved it between the 4 and 8.  That looks like a move of 8 places.

What direction did it move?
Looks like we moved it to the left.

So, our count is +8.

 

 
example 7b

 
Note how the number we started with is a bigger number than the one we are multiplying by in the scientific notation.  When that is the case, we will end up with a positive exponent

 
 
notebook Example 8:   Write the number in scientific notation:   .00054.

 

 
example 8a
*Decimal is at the beginning of the number

*Move decimal to create a number between 1 and 10

 

 
How many decimal places did we end up moving?
We started at the beginning of the number .00054  moved it between the 5 and 4.  That looks like a move of 4 places.

What direction did it move?
Looks like we moved it to the right.

So, our count is - 4.

 

 
example 8b

 
Note how the number we started with is a smaller number than the one we are multiplying by in the scientific notation.  When that is the case we will end up with a negative exponent.

 
 
Write a Scientific Number in Standard Form

 
Basically, you just multiply the first number times the power of 10. 

Whenever you multiply by a power of 10, in essence what you are doing is moving your decimal place.

If the power on 10 is positive, you move the decimal place that many units to the right.

If the power on 10 is negative, you move the decimal place that many units to the left.

Make sure you add in any zeros that are needed 

 
notebook Example 9:  Write the number in standard notation, without exponents.
example 9a

 
example 9b

*Move the decimal 6 to the right

 
 
notebook Example 10:  Write the number in standard notation, without exponents.
example 10a

 
example 10b

*Move the decimal 5 to the left

 

 

desk Practice Problems

 
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1b: Simplify, use positive exponents to write each answer.

 


 

pencilPractice Problem 2a: Write the number in scientific notation.

 

2a.      .00000146
(answer/discussion to 2a)

  


 

pencil Practice Problem 3a: Write the number in standard notation, without exponents.

 


 

 

 

desk Need Extra Help on these Topics?

 

The following are webpages that can assist you in the topics that were covered on this page: 
 

http://www.sosmath.com/algebra/logs/log3/log32/log32.html
This webpage helps with the quotient rule for exponents.

http://www.purplemath.com/modules/exponent.htm
This webpage gives an overall review of exponents.  It contains rules from both this tutorial and Tutorial 26: Exponents.


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

 

 

 

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Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.